Here’s how to construct the figure based on your instructions:

Steps for the construction:

1. Draw a line segment MN of 5 cm:

   • Use a ruler and draw a straight line segment of length 5 cm. Label the two endpoints as $M$ and $N$.

2. Construct an angle of 60° at point M:

   • Place the compass on point $M$, and draw an arc of any radius that intersects line segment $MN$.
   • Without changing the radius of the compass, place the compass on the point where the arc intersects $MN$, and draw another arc.
   • From $M$, draw a straight line through the intersection of the arcs. This will give you a 60° angle. Label this new ray as $MP$.

3. Construct the perpendicular at point N:

   • To construct a perpendicular to line segment $MN$ at point $N$, place the compass at point $N$ and draw an arc that crosses $MN$ at two points (on either side of $N$).
   • Without changing the compass width, place the compass at each of the two points where the arc intersects $MN$ and draw two more arcs that intersect each other above $N$.
   • Draw a straight line through $N$ and the intersection of the arcs above $N$. This is the perpendicular.

4. Mark the point P:

   • Extend the ray $MP$ so that it intersects the perpendicular line you just constructed.
   • Mark the intersection point as $P$.

Now, you should have:

• A line segment $MN$ of 5 cm.
• A 60° angle at point $M$ with $MN$ as one of its arms.
• A perpendicular line at point $N$, and the point $P$ where the other side of the 60° angle intersects the perpendicular.

Would you like further details on any step or how to verify these constructions?

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Related Questions:

1. How can we verify that the angle constructed is exactly 60°?
2. What are alternative methods to construct a perpendicular at point $N$?
3. How can we check that point $P$ is correctly placed?
4. What is the geometrical significance of the point $P$ in this construction?
5. How would you modify the construction if the angle at $M$ was different, say 45°?

Tip: Always use a sharp pencil and accurate instruments (compass and ruler) to make precise constructions.